Interpolation (Mathematics)
Interpolation is a mathematical technique used to estimate unknown values based on known data points within a sequence. It serves as a method for predicting values that lie between established data points, contrasting with extrapolation, which seeks to determine values outside a known range. Among the various forms of interpolation, linear interpolation is the most straightforward, using two known points to calculate an intermediate value along a straight line. For example, if one point is 4 and another is 8, the interpolated value would be 6.
In addition to linear interpolation, cubic interpolation improves accuracy by considering four data points instead of two, which helps to better accommodate non-linear data patterns. Logarithmic interpolation is another variation that deals with values that increase exponentially, providing another method for estimating unknown values based on a different growth model. Interpolation finds practical applications in various fields, including finance, meteorology, and event planning, where it helps to make informed estimates based on existing data. Despite its usefulness, linear interpolation relies on the assumption that the data follows a predictable pattern, making it important to choose the appropriate method based on the context of the data being analyzed.
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Interpolation (Mathematics)
Interpolation is a method used in mathematics to estimate an unknown value between two known values in a sequence. Several types of interpolation exist; among the most common are linear, cubic, and logarithmic. Interpolation is the opposite of extrapolation, which is a method of determining values beyond a known value range. The names of both terms reflect their functions, with the Latin prefix inter- meaning "between" and extra- meaning "outside."
Overview
The simplest and most common method of interpolation is linear interpolation. This can be illustrated by representing the known numerical values as two data points on a straight line. For example, if one value is known to be 8 and the other is 4, then the unknown value (x) can be estimated to be the data point directly between the two values, or 6. Another way to do this is to list a series of known data points and find the unknown value by calculating the relationship between the points. For instance, consider this set of values:
20 = 100
40 = x
80 = 700
Since the first and third values in the right column increase by 600 and the first and third values in the left column increase by 60, it can be determined that an increase by 10 in the left column is equal to an increase by 100 in the right column. Therefore the value of x is equal to 300.
In practical applications, linear interpolation can be used to determine calculations such as interest rate payments on a half-month basis. Because many interest rates are determined on a one- or two-month basis, these values are known. Using interpolation, a financial analyst can estimate the rate for a period that falls within that range. It may be used by meteorologists to forecast temperatures at a location between a cold front and a mass of warmer air or by caterers trying to determine how much food to prepare for a dinner party when the number of people attending unexpectedly drops.
While linear interpolation is a good way to estimate an unknown value, it assumes that the observed data points continue on an easily predictable, linear path. It also assumes that the data model used to arrive at the values is correct. Of the many other types of interpolation, cubic interpolation is one simple way to fix that issue. Cubic interpolation adds two more data points to the equation on either side of the two original points. Instead of two points being used to find an unknown value, four are used. Logarithmic interpolation uses values that increase in an exponential rather than linear manner. This means that base values increase according to a numerical power. For example, 102—or 10 increased to the power of 2—is the equivalent of 10 times 10, or 100.
Bibliography
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